Localization algorithms for asynchronous time difference of arrival positioning systems
 Shuai He^{1}Email author,
 Xiaodai Dong^{1} and
 WuSheng Lu^{1}
https://doi.org/10.1186/s1363801708511
© The Author(s) 2017
Received: 10 June 2016
Accepted: 21 March 2017
Published: 11 April 2017
Abstract
An asynchronous time difference of arrival (ATDOA) positioning system requires no time synchronization among all the anchor and target nodes, which makes it highly practical and can be easily deployed. This paper first presents an ATDOA localization model, and then primarily focuses on two new localization algorithms for the system. The first algorithm is a twostep positioning algorithm that combines semidefinite programming (SDP) with a Taylor series method to achieve global convergence as well as superior estimation accuracy, and the second algorithm is a constrained leastsquares method that has the advantage of low complexity and fast convergence while maintaining good performance. In addition, a novel receiver reselection method is presented to significantly improve estimation accuracy. In this paper, we also derive the CramerRao lower bound (CRLB) of the ATDOA positioning system using a distancedependent noise variance model, which describes a realistic indoor propagation channel.
Keywords
1 Introduction
Position information brings enormous benefits to many reallife applications ranging from cargo tracking, tourist guiding, emergency evacuation, to countless usage scenarios. As mobile devices become ubiquitous, contextual awareness applications have become popular, and the indoor positioning system has gained significant attention. The timebased localization method, including oneway time of arrival (TOA) and time difference of arrival (TDOA), exploits the fine delay resolution property of wideband signals and has great potential for providing high accuracy location estimation. However, both methods face a major challenge, that is, synchronization is required among the clocks of the involved nodes with a timing accuracy proportional to the desired localization precision.
Efforts have been made in the literature to relax the synchronization requirements, and two common methods are twoway ranging [1, 2] and elliptical localization [3–5]. In twoway ranging, an anchor node transmits a packet to a target node, which replies by an acknowledgment packet to the anchor node after a response delay. The twoway ranging eliminates the error due to imperfect synchronization between nodes, yet this approach is sensitive to clock nonidealities [3, 6]. The elliptical localization system starts with an anchor transmitter (Tx) emitting a pulse, and upon arrival, the pulse is retransmitted by the target node. An anchor receiver (Rx) captures two pulses in a row, one from the anchor Tx and the other from the target. The time difference between the two received signals can be measured, and together with the knowledge of the anchor Tx and Rx positions, the sum of the distances between the target and the two anchor nodes can be calculated. Hence, the target node lies on the trajectory of an ellipse with anchor Tx and anchor Rx as the two foci. Several elliptical localization systems have been studied in the literature [3–5]. These systems work in a similar manner, and they differ in one or two respects. The system deployment in [3] has a designated anchor Tx emitting an ultrawideband pulse, and three anchor Rx nodes to perform the time difference arrival measurements. Wang et al. [4] proposed an asymmetric trip ranging protocol, and the system deployment is similar to [3], but it involves a timing logic at the target node, which suffers from clock nonidealities. In [5], a distributed localization scheme is proposed, and it uses the target node to measure the TDOA. Due to the cost and power constraints on the target node, lowperformance clocks are normally employed which limits the accuracy. In this paper, we first present a new elliptical localization system, namely, an asynchronous time difference of arrival (ATDOA) positioning system. The ATDOA system’s deployment is different from [3] in that there is no need for a designated anchor Tx. Rather, the proposed simplest deployment contains one anchor Rx and three anchor Tx. A more comprehensive setup contains four transceiver anchors, each of which can be dynamically configured into a Tx or Rx in order to minimize estimation error by performing novel receiver reselection. More importantly, two new location estimation algorithms tailored for the ATDOA system are proposed and studied. More details can be found at [7].
Due to the imperfect implementation of location sensing systems, lack of bandwidth, added thermal noise, multipath of the radio propagation channel, and the drift of the clocks, there are always errors associated with measurements of location related metrics. To obtain an estimate of target location in the presence of measurement errors, a variety of direct and iterative positioning algorithms have been developed. When measurement error distribution is available, a maximum likelihood estimator (MLE) is commonly used. An approximate maximum likelihood (ML) algorithm was developed in [8] to achieve nearoptimal performance without the complexity of “full” maximum likelihood estimation. In [9], a MLbased algorithm was proposed, and simulation results reveal that the solution closely approaches the fundamental bounds. In spite of attaining optimum estimation performance, the ML approach requires sufficiently precise initial estimates for global convergence. In [10], it has been shown that the positioning accuracy of the ML methodology attains CramerRao lower bound (CRLB) at sufficiently small noise conditions. However, it is difficult to implement in practice because the ML cost function contains multiple local minima and maxima, hence, its maximization is sensitive to initial conditions, and there is no guarantee of global optimality [11]. In [12], results show that even when the ML estimator is initialized by a weighted least squares estimate, which is close to the global solution, it still converges occasionally to a local minimum. Unlike the ML approach, the least squares (LS) approach does not assume any characterization of the noise statistic affecting the observations; hence, it is deemed a suboptimal method [13]. However, it has low computational complexity and therefore is easy to implement in a practical system. Basically, there are two approaches for solving the nonlinear LS equations. The first approach is to solve them directly in a nonlinear least squares (NLS) or weighted least squares (WLS) framework [14–16]. The common procedure is linearization followed by gradient searches. Although optimum estimation performance can be attained, it requires sufficiently precise initial estimates for global convergence because the corresponding cost functions are multimodal. The second approach is linear least squares (LLS) method. It reorganizes the nonlinear equations into a set of linear equations so that realtime implementation is allowed, and global convergence is ensured [11, 17–21].
Although the MLE has the highest accuracy, it is highly nonlinear and does not assure global convergence. It is possible to relax the ML formulation to a semidefinite programming (SDP) problem in order to provide a highfidelity approximate solution that can be obtained in a globally optimum fashion with reduced computational efforts. Hence, we first develop a twostep algorithm that takes advantage of the SDP’s global convergence property to provide a solution, that is then used as an initial estimate for a Taylor series method to achieve superior accuracy. The twostep method provides accurate solutions at a cost of considerable computational complexity, and it may not be an ideal approach for applications where computational resources are limited. Therefore, we also present a constrained leastsquares (CLS) estimator that provides good solution accuracy with reduced complexity for ATDOA positioning systems.

A practical CRLB has been derived for the ATDOA system. We model the received signal’s signaltonoise ratio (SNR) as a distancedependent parameter to derive a more accurate and a more practical lower bound.

A twostep (SDP + Taylor) algorithm and a CLS algorithm are proposed to estimate the target position in the ATDOA system. The twostep estimator can be applied in applications where accuracy is the most critical, and the CLS estimator is very useful in realtime systems and mobile devices where battery life and computational capability is limited.

The localization algorithm’s performance is thoroughly studied based on practical achievable ranging accuracy. This unique analysis method allows us to fully understand the advantages and disadvantages of different algorithms.
We follow the standard terminology in the literature to call the nodes with known positions anchors and the node to be localized the target node. Bold upper case symbols denote matrices and bold lower case symbols denote vectors. The 0 _{ m×n } is the m×n zero matrix and I _{ m } is the m×m identity matrix. The transpose and 2norm of a vector x are denoted by (·)^{ T } and ∥x∥, respectively. For two symmetric matrices A and B, A≽B means that A − B is positive semidefinite.
2 System model
An ATDOA localization system consists of a number of nodes. The anchor node that initiates the pulse transmission is called anchor Tx, and the one that receives the pulse is called anchor Rx. In an ATDOA system, there are multiple anchor Tx nodes and one anchor Rx node.
Let x=[x,y]^{ T } and x _{ i }=[x _{ i },y _{ i }]^{ T },i=1,2,…,M be the coordinates of the target node and anchor nodes, respectively, where M is the number of anchor nodes with M≥3 for twodimensional positioning. Without loss of generality, let anchor Rx be at x _{1}, and anchor Tx be positioned at x _{ i }, i=2,3,…,M.
where n _{ i } is a zero mean measurement error. Equation (1) exhibits the beauty of the ATDOA system that the time difference (t _{ ARR }−t _{ ARD }) is measured at and only at anchor Rx. Therefore, no clock synchronization is required among anchor Rx, anchor Tx and the target node. The use of the backbone cables which are mandatory in conventional TDOA positioning systems can now be avoided.
3 CramerRao lower bound
CramerRao lower bound is commonly used for providing a lower bound on an estimator’s mean square error (MSE). It establishes a fundamental limit on the achievable localization accuracy, and it serves as a benchmark for any unbiased location estimator. Previous works derive CRLB based on modeling range estimates as being corrupted by zero mean Gaussian noise [22, 23]. These works made an assumption that the variance of the range estimate is not dependent on the actual node pair distance. As a matter of fact, signal power decays as the propagation distance increases in practical situation. In an indoor environment, the path loss exponent can vary from 2 to 6 [24], and the signal power decays 20 to 60 dB as the propagation distance increases by a decade. This results in a significantly received signal power variation. Given a constant thermal noise level, received signal power variation results in a change in SNR, which in turn determines the achievable localization accuracy [23]. To reflect the SNR change, we follow a similar approached used in [25] to model noise variance as a distance dependent parameter. Such modeling is applied throughout this paper. Below, we derive a distancedependent CRLB for the ATDOA system.
Typically, the constant K _{ E } is extremely small and therefore the third terms in (11), i.e., \(\frac {{{f_{xxi}}}}{{{K_{E}}{g_{i}}}}\), \(\frac {{{f_{yyi}}}}{{{K_{E}}{g_{i}}}}\) and \(\frac {{{f_{xyi}}}}{{{K_{E}}{g_{i}}}}\) dominates.
4 A high accuracy twostep localization algorithm
In this section, we propose a twostep localization algorithm that combines a SDP technique and a Taylor series method to achieve high estimation accuracy. Typically, SDP is used to relax the nonconvex problem to a convex problem so as to obtain a global estimation of the true position regardless of the initial point used [27]. The solution is then used as an initial guess for the Taylor series method to achieve superior performance. In addition, as the SDP method can achieve global minimal, if necessary, the estimated target position can be used to reselect the anchor Rx node to minimize the estimation error. Below, we derive the twostep localization algorithm.
In minimizing the objective function in (14a), h _{ i1} tends to decrease while h _{ i } and h _{1} tend to increase, hence, the relaxation made above is not tight. Nevertheless, (14) is a convex problem whose global solution can readily be computed. In addition, simulation studies have indicated that the approximate solution to problem (14) is typically close to the true location. Based on these, we propose a twostep algorithm in that the SDP solution serves as an initial estimation to allow a Taylorseriesbased method step a quick convergence to an accurate location estimation.
Let x _{0}= [ x _{0},y _{0}]^{ T } be the initial guess of the target location and Δ x= [ δ x,δ y]^{ T } be the small increment on x.
The updated target location is utilized in the next iteration until the magnitude of Δ x becomes less than a prescribed tolerance. It is reasonable to treat the measurement error variance \(\sigma _{i}^{2}\) as a known value in both SDP and Taylor steps, because modern receiver is capable of measuring signaltonoise ratio which is inversely related to the \(\sigma _{i}^{2}\). Simulation results and analysis of the twostep estimator are provided in Section 6.
5 A low complexity constrained leastsquares localization algorithm
with t _{ i }=∥x−x _{ i }∥, q _{ i }=r _{ i }+∥x _{ i }−x _{1}∥ and r _{ i } representing the measured range differences.
We remark that matrix B in (22b) is independent of measurements, hence, V _{1} , S , U and v _{ M } can be precalculated; and for twodimensional location problems H(x ^{ k }) is of size 3×3, hence, the complexity of computing H ^{−1}(x ^{ k }) as required in (32) is insignificant. The algorithm is found insensitive to its initial point \([\mathbf {x}_{0}^{T} \; \phi _{0}]^{T}\) as long as it is a reasonable one, e.g., \(\mathbf {x}_{0} = \frac {1}{M}\sum _{i=1}^{M} \mathbf {x}_{i}\) and ϕ _{0}=0. Typically, the algorithm converges in less than five iterations. Simulation results of the CLS algorithm and a detailed comparison with other estimators are presented in Section 6.
6 Simulation results
Computer simulations have been conducted to corroborate the theoretical development and to evaluate the performance of the twostep and the CLS estimators. Four algorithms, namely, the twostep algorithm, the CLS algorithm, the linear least squares algorithm [11], as well as the SDP algorithm are compared. In addition, a comparison to the CRLB is provided to showcase the great accuracy achieved.
We adopted a consistent system geometry as shown in Section 2, with four anchor nodes placed at the vertex of a square, i.e., at (0, 0) m, (0, 100) m, (100, 100) m, and (100, 0) m. To fully evaluate the performance of the estimators, the target node is set to sweep a 100×100 m grid with a step size of 1 m moving towards either X or Y direction. The starting location is (0, 0) m, and the stopping location is (100, 100) m. To solve the SDP problem involved, the convex solver CVX [30] is applied. The initial guess point of the CLS algorithm is set to the mean value of the anchor nodes coordinate \(\frac {1}{4} \sum _{k=1}^{4} \mathbf {x}_{k}\). Receiver reselection technique is applied in all simulations to achieve the best possible performance. MSE is employed as the performance measure.

Low ranging error (σ _{0}≤0.1m): the ranging error is within ±0.2 m in 95% of the time. The relative error percentage is \(P_{e} = \frac {0.1}{50\sqrt {2}} \cdot 100\%= 0.14\% \). Such high ranging accuracy is rarely reported in literature. It was only achieved in very carefully controlled experiment environments where high cost and high precision lab instruments were employed [31–33].

Medium ranging error (0.1 m<σ _{0}<10 m): the relative error percentage is within 0.14 to 14%. Most published works using TOA, TDOA, and twoway ranging techniques fall within this range [34–39]. The ATDOA system belongs to this category as well.

High ranging error (σ _{0}≥10 m): the relative error percentage is greater than 14%. Many system employing RSS ranging method fits in this category [40–42].
To the best of the author’s knowledge, there were no other works that thoroughly study the localization algorithm performance according to practical achievable ranging accuracy. This analysis method allows us to fully understand the advantages and disadvantages of each estimators, and hence, is of great importance to guide the selection of the algorithms in a reallife system. From Section 6.1 to Section 6.3, simulated performance of each algorithms under the aforementioned three scenarios are presented. Section 6.4 provides a comparison of algorithms with varying error magnitudes.
6.1 Low ranging error simulation results
This section presents simulation results with an error standard deviation of 0.1 m in a 100×100 m area. The relative error percentage is extremely low, and such scenario is not very common in practical systems. Nevertheless, it well represents a system with a very high signal to noise ratio.
Summary of the simulated MSE with a ranging error of 0.1 m
Low ranging error simulation results summary  

Estimators  Mean (dB)  STD (dB)  Maximum (dB)  Minimum (dB) 
LLS  −3.1  0.6  7.6  −8.1 
SDP  −2.2  0.5  4.5  −9.9 
CLS  −5.1  0.5  3.7  −9.1 
Twostep  −10.1  0.3  −5.6  −12.3 
It is very obvious in Fig. 5 that the twostep (SDP + Taylor) estimator outperforms all the other estimators. The average MSE over the entire 100×100m grid is several dB lower. In addition, the twostep estimator’s performance is rather consistent across the entire area. The CLS estimator has the second lowest MSE in all four estimators. Its average MSE is 5 dB higher than the twostep estimator, but 2 dB lower than the LLS estimator. Hence, the CLS estimator is a good compromise between the need for high accuracy and the demands of low complexity. The LLS estimator’s performance is reasonably satisfactory given it has the lowest complexity and an analytical solution. The SDP estimator performs the worst when the ranging error is low, yet we will find in Section 6.3 that it outperforms all the other algorithms when the ranging error is high. Another observation is that the MSE on the square edge is significantly higher than other positions for LLS, CLS, and the twostep estimators, and this is consistent with the CRLB shown in Section 3.
6.2 Medium ranging error simulation results
This section presents simulation results with an error standard deviation of 1 m in a 100×100 m area. It well represents a practical system using time based localization techniques such as TOA, TDOA, and ATDOA.
Summary of the simulated MSE with a ranging error of 1 m
Medium ranging error simulation results summary  

Estimators  Mean (dB)  STD (dB)  Maximum (dB)  Minimum (dB) 
LLS  16.8  0.6  28.3  12.0 
SDP  11.8  0.2  15.9  7.4 
CLS  17.2  1.0  24.3  10.2 
Twostep  10.0  0.2  14.3  6.9 
Evidently, the twostep estimator still outperforms all the others and is still robust regardless of the target location. The SDP estimator performs the worst in low ranging error condition, however, its superiority is convincingly demonstrated as the error standard deviation increases to 1 m. There is only less than 2 dB difference between the SDP and the twostep estimator. The average MSE of the LLS, and the CLS estimators are comparable. The CLS provides a more accurate estimation in the center, while the LLS is generally better on the edge.
6.3 High ranging error simulation results
This section presents simulation results with an error standard deviation of 10 m in a 100×100 m area, to study each estimators’ performance in a high relative error percentage condition, i.e., P _{ e }>14%. Although the ATDOA system generally has less than 14% relative percentage error, it is still worthwhile to study its performance under high ranging error condition. That is because the wireless channel varies with high dynamic range by shadowing and fading effects, which can cause the ranging accuracy to change significantly.
Summary of the simulated MSE with a ranging error of 10 m
Medium ranging error simulation results summary  

Estimator  Mean (dB)  STD (dB)  Maximum (dB)  Minimum (dB) 
LLS  34.8  1.2  65.2  30.4 
SDP  29.4  0.7  34.0  21.5 
CLS  33.1  0.4  35.9  28.7 
Twostep  30.0  0.3  38.6  26.9 
As ranging error increases to 10 m, the SDP estimator’s average error is 0.6 dB less than the twostep estimator, and becomes the most accurate among all estimators. Although the twostep estimator is not the best performed under high ranging error condition, it still performs satisfactorily well. The CLS estimator performs consistently well regardless of the ranging error level. It has a low variation across the 100×100 m area, showcasing its strong robustness. The LLS estimator does not work well in high error condition. Its lowest estimation error is comparable to the maximum estimation error of the other estimators. Besides, its estimation error is particularly dependent on the target location.
6.4 Estimation accuracy versus ranging error
Extensive simulations have been conducted to evaluate the performance of the twostep and the CLS algorithm under varying ranging errors and to compare their performance against the LLS, SDP, and the CRLB. Unlike Section 6.1 to Section 6.3, we fix the target node location and vary the ranging error magnitude, so we can compare their performance from a different angle.
6.5 Algorithm complexity
It is well known that the complexity of solving the above system of equations is about M ^{3}/3. Since the algorithm needs to solve the SDP problem (14) only once plus K iterations in step two, the complexity of the algorithm in Section 4 is in the order of O(M ^{5})+K M ^{3}/3.
for d _{ k }. We remark that above complexity analysis for the CLS method does not take the SVD of matrix B into account because B is a constant matrix (see Eq. (22b)) whose SVD can be performed offline before the system starts to operate.
7 Conclusions
An ATDOA positioning system and two associated location estimation algorithms are presented in this paper. The distinct advantage of the ATDOA system is that no clock synchronization is needed. Therefore, the complexity of the system can be reduced significantly. Besides, by properly selecting the anchor Rx node, the ATDOA system can achieve superior performance. In practice, as noise variance is dependent on the ranging distance, we have adopted a distancedependent noise model to derive CRLB and to conduct simulations.
More importantly, two new localization algorithms, namely, the twostep and constrained LS algorithms have been proposed to provide position estimation in the ATDOA system. The twostep estimator combines the SDP and Taylor series methods to achieve global convergence and superior estimation accuracy. The constrained LS algorithm obtains good performance while keeps the computational complexity low, and the convergence speed is fast. Simulation results indicate that both estimators are able to achieve great performance regardless of the measurement error level. For the timebased localization systems, such as TOA, TDOA, ATDOA, and so on, the ranging error is relatively low, and under this condition, the twostep estimator achieves the best accuracy. In addition, its estimation accuracy is quite consistent regardless of the target node location and ranging error. Therefore, it can be applied in applications where accuracy is the most critical. The CLS estimator’s performance is slightly worse than the twostep estimator, nevertheless, it consumes less CPU time and requires lower computational complexity. Hence, it is very useful in realtime systems and mobile devices where battery life and computational capability is limited. In this regard, these two algorithms may be considered as a complementary pair of solution tools that provide the system designer with more than one option for an appropriate tradeoff between accuracy and complexity.
8 Appendix. proof of result (14)
Furthermore, by introducing a new variable z=x ^{ T } x to linearize constraint (39c), the above problem can be relaxed to a standard SDP problem which yields the result of (14).
Declarations
Funding
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant 261524.
Authors’ contributions
SH and XD developed the system model and the CR lower bound. SH and WSL formulated the optimization algorithms, SH performed simulation, and all authors contributed to the interpretation of the results and writing of the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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